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Theorem isoml 39239
Description: The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)
Hypotheses
Ref Expression
isoml.b 𝐵 = (Base‘𝐾)
isoml.l = (le‘𝐾)
isoml.j = (join‘𝐾)
isoml.m = (meet‘𝐾)
isoml.o = (oc‘𝐾)
Assertion
Ref Expression
isoml (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦
Allowed substitution hints:   (𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)   (𝑥,𝑦)

Proof of Theorem isoml
Dummy variable 𝑘 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6906 . . . 4 (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2 isoml.b . . . 4 𝐵 = (Base‘𝐾)
31, 2eqtr4di 2795 . . 3 (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4 fveq2 6906 . . . . . . 7 (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5 isoml.l . . . . . . 7 = (le‘𝐾)
64, 5eqtr4di 2795 . . . . . 6 (𝑘 = 𝐾 → (le‘𝑘) = )
76breqd 5154 . . . . 5 (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦𝑥 𝑦))
8 fveq2 6906 . . . . . . . 8 (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
9 isoml.j . . . . . . . 8 = (join‘𝐾)
108, 9eqtr4di 2795 . . . . . . 7 (𝑘 = 𝐾 → (join‘𝑘) = )
11 eqidd 2738 . . . . . . 7 (𝑘 = 𝐾𝑥 = 𝑥)
12 fveq2 6906 . . . . . . . . 9 (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
13 isoml.m . . . . . . . . 9 = (meet‘𝐾)
1412, 13eqtr4di 2795 . . . . . . . 8 (𝑘 = 𝐾 → (meet‘𝑘) = )
15 eqidd 2738 . . . . . . . 8 (𝑘 = 𝐾𝑦 = 𝑦)
16 fveq2 6906 . . . . . . . . . 10 (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
17 isoml.o . . . . . . . . . 10 = (oc‘𝐾)
1816, 17eqtr4di 2795 . . . . . . . . 9 (𝑘 = 𝐾 → (oc‘𝑘) = )
1918fveq1d 6908 . . . . . . . 8 (𝑘 = 𝐾 → ((oc‘𝑘)‘𝑥) = ( 𝑥))
2014, 15, 19oveq123d 7452 . . . . . . 7 (𝑘 = 𝐾 → (𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)) = (𝑦 ( 𝑥)))
2110, 11, 20oveq123d 7452 . . . . . 6 (𝑘 = 𝐾 → (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) = (𝑥 (𝑦 ( 𝑥))))
2221eqeq2d 2748 . . . . 5 (𝑘 = 𝐾 → (𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) ↔ 𝑦 = (𝑥 (𝑦 ( 𝑥)))))
237, 22imbi12d 344 . . . 4 (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
243, 23raleqbidv 3346 . . 3 (𝑘 = 𝐾 → (∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
253, 24raleqbidv 3346 . 2 (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
26 df-oml 39180 . 2 OML = {𝑘 ∈ OL ∣ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))))}
2725, 26elrab2 3695 1 (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 𝑦𝑦 = (𝑥 (𝑦 ( 𝑥))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061   class class class wbr 5143  cfv 6561  (class class class)co 7431  Basecbs 17247  lecple 17304  occoc 17305  joincjn 18357  meetcmee 18358  OLcol 39175  OMLcoml 39176
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-oml 39180
This theorem is referenced by:  isomliN  39240  omlol  39241  omllaw  39244
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